The Rudolf Steiner Archive

12. Formal Logic I

[ 1 ] We have already touched upon the relationship between anthroposophy and philosophy, as it were, in passing; today we want to discuss some fairly elementary aspects of what is known as formal logic. Despite the elementary nature of our discussion today, it may not be without value to include a philosophical chapter among our glimpses into higher worlds. This is not to say that such a lecture could directly contribute to penetrating the higher worlds. A logical examination can do no more in this regard than formal logic can enrich experience in the sensory realm. For example, someone who has never seen a whale cannot be convinced of its existence through proof alone. They must make the observation themselves. But it is precisely the understanding of these borderlands that will be useful to anthroposophy, just as logic, for example, was useful to the Scholastics. The philosophers of the Middle Ages—who are today somewhat disparagingly grouped under the name “Scholastics”—did not regard logic as an end in itself; it did not serve to teach any substantive content. Rather, the subject matter—the content of their teaching—was either sensory observation or revelation, which is attained through divine grace. But although, in the opinion of the Scholastics, logic was entirely powerless to enrich experience, they nevertheless regarded it as an important instrument of defense. So it should also be such an instrument for us.

[ 2 ] A distinction is made between material and formal logic. Logic as such cannot even conceive of something material or substantive as its object. Concepts such as time, number, and God provide a substance that does not arise from logical inferences. In contrast, the form of thought is the task of logic; it brings order to our thoughts and teaches us how to connect concepts in a way that leads to correct conclusions. It is fair to say that logic was more highly valued in earlier times than it is today. In high schools, philosophical propaedeutics, logic, and psychology were once taught. The instruction aimed to guide young people toward disciplined, orderly thinking; “propaedeutics” means “preparation.” Today, there is a drive to eliminate this entire form of preparation and incorporate it into the study of grammar, because logic is no longer given sufficient respect. Thinking, it is said, is innate to human beings; so why teach thinking at all in a separate branch of instruction? But especially in our time, it is very necessary to practice self-reflection in this area and to engage more deeply with formal logic again.

[ 3 ] Aristotle is the founder of formal logic. And Aristotle’s contributions to logic have always been recognized, even by Kant, who states that formal logic has not advanced much since Aristotle. More recent thinkers have sought to add to it. We will not examine today whether such additions were necessary and justified or not. We need only recognize the scope of logic here.

[ 4 ] Anthroposophists are often accused of not being logical. This is very often because the person making the accusation does not know at all what logical thinking is or what the laws of logical thinking are. Logic is the study of the correct, harmonious connection of our concepts. It encompasses the laws by which we must govern our thoughts so that we may have within ourselves a mirror of the true relationships of reality.

[ 5 ] First, we must clarify what a concept is. The fact that people have such a vague understanding of what a concept is is, once again, due to the lack of in-depth study of logic on the part of the scholarly community. When we encounter an object, the first thing that occurs is perception. We notice a color, a taste, or a smell, and this phenomenon—which takes place between the person and the object—we must initially regard as characterized by sensation. What is contained in the statement “Something is warm, cold, and so on” is a sensation. However, in everyday life, we do not actually experience this pure sensation at all. When we look at a red rose, we do not merely perceive the color red; rather, whenever we interact with objects, we immediately experience a cluster of sensations. We call the combination of the sensations “red, scent, size, and shape” a “rose.” We do not actually have individual sensations, but only clusters of sensations. Such a cluster can be called a “perception.”

[ 6 ] In formal logic, one must make a sharp distinction between perception and sensation. Perception and sensation are entirely different things. Perception is the first thing that presents itself to us; it must first be analyzed before we can have a sensation. However, what gives us the content of our mind is not the only thing. The rose, for example, makes an impression on us: its red color, scent, shape, and size. When we turn away from the rose, we retain something in our mind—a faded remnant of the red color, the scent, the size, and so on. This faded remnant is the representation. One should not confuse perception with representation. The representation of a thing is that in which the thing itself is no longer present. The representation is already a memory image of the perception.

[ 7 ] But we still haven’t arrived at the concept. We obtain the representation by exposing ourselves to impressions from the external world. We then retain the concept as an image. Most people never go beyond the concept throughout their lives; they do not advance to the actual concept. What a concept is and how it relates to a concept is best illustrated by an example from mathematics. Let’s take the circle. If we sail out to sea in a boat until we reach a point where we see nothing but the surface of the sea and the sky, we can, when it is completely calm, perceive the horizon as a circle. If we then close our eyes, we retain the representation of the circle as a mental image from this perception. To arrive at the concept of the circle, we must take a different path. We must not seek an external source for this concept; rather, we construct in our minds all the points on a surface that are equidistant from a specific fixed point; if we repeat this countless times and mentally connect these points with a line, the image of a circle takes shape in our minds. We can also illustrate this mental image with chalk on the blackboard. If we now visualize this image of the circle—which arose not from external impressions but through internal construction—and compare it with the image of the sea’s surface and the horizon that presented itself to external perception, we may find that the internally constructed circle corresponds perfectly to the image of external perception.

[ 8 ] If people truly think logically—in the strict logical sense—then they are doing something other than perceiving external reality and recalling what they have perceived; that is merely a mental image. In logical thinking, however, every thought must be constructed internally; it must be created in a manner similar to what I have just explained using the example of the circle. Only then, with this inner mental image, does a person approach external reality and find harmony between the inner image and external reality. The mental image is connected to external perception; the concept arises through internal construction. People who truly thought logically have always constructed ideas internally in this way. Thus, when Kepler formulated his laws, he constructed them internally, and he then found them to be in harmony with external reality.

[ 9 ] The concept, then, is nothing other than a mental image; it has its genesis, its origin, in thought. An external illustration is merely a crutch, an aid to making the concept more vivid. The concept is not derived from external perception; it exists, at first, only in pure interiority.

[ 10 ] In terms of its thinking—with the exception of mathematics—our intellectual culture today has not yet progressed beyond mere imagination. For the researcher of the human spirit, it is sometimes grotesque to see how little people have progressed beyond mere imagination. People generally believe that a concept originates from imagination and is merely a paler, less substantial version of it. For example, they believe they arrive at the concept of a horse by seeing large, small, brown, white, and black horses appear one after another in their perception; and now—so people continue to reason—I extract from the perception of these various horses what all horses have in common and set aside what distinguishes them, and in this way I arrive at the concept of a horse. - But in this way one obtains only an abstract idea; one never, in the strict sense of the word, arrives at the concept of a horse. Nor does one arrive at a concept of a triangle by taking all kinds of triangles, extracting what they have in common, and omitting what distinguishes them. One arrives at a concept of the triangle only by mentally constructing the figure of three intersecting lines. With this mentally constructed concept, we approach the external triangle and find that it harmonizes with the image we have constructed in our minds.

[ 11 ] It is only in regard to mathematical concepts that people in our present-day culture are able to rise to the level of the concept. For example, one proves through internal construction that the sum of the angles in a triangle is equal to one hundred eighty degrees. But if someone were to begin constructing concepts of other things through inner construction as well, a large portion of our philosophers would not recognize this at all. Goethe created the concepts of the “primordial plant” and the “primordial animal” through inner construction; it was not merely a matter of omitting the differences and retaining the similarities—as was mentioned earlier in the example of the horse. The “primordial plant” and the “primordial animal” are such inner mental constructs. But how few people recognize this today. Only when one can build up the concept of the horse, the plant, the triangle, and so on through inner construction—and when this corresponds to external perception—only then does one arrive at the concept of a thing. Most people today hardly know what is meant when one speaks of conceptual thinking.

[ 12 ] Let us not consider mathematical concepts, nor Goethe’s concept of organicity—where he created concepts in a truly magnificent way—but let us instead consider the concept of virtue. One can, of course, have a vague, general idea of virtue. But if one wants to arrive at a concept of virtue, one must construct it internally, and one must draw on the concept of individuality. One must construct the concept of virtue in the same way that one constructs the concept of a circle. This requires some effort, and various elements must be brought together, but it is just as possible as constructing mathematical concepts. Moral philosophers have always attempted to provide a concept of virtue free from sensuality. Some time ago, there was a philosopher who could not conceive of a concept of virtue free from sensuality and who regarded those who claimed such a thing as fantasists. He explained that when he thought of virtue, he imagined it as a beautiful woman. Thus, he introduced a sensual element into the non-sensual concept. And because he could not conceive of a concept of virtue free from sensuality, he denied that others could do so either.

[ 13 ] If you delve into Herbart’s ethics, you will find that, for him, “goodwill” and “freedom”—these ethical concepts—are not formed by taking what they have in common and omitting what divides them; rather, he says, for example, that benevolence encompasses the relationship between one’s own volitional impulses and the imagined volitional impulses of another person. - He thus provides a pure conceptual definition. In this way, one could construct the entire system of morality using pure concepts, much like mathematics, and as Goethe attempted to do with his “Organic Theory.” The general notion of virtue must therefore not be confused with the concept of virtue. People arrive at this concept gradually through a process of inner construction.

[ 14 ] By setting the concept of the concept before us, we distance ourselves from everything arbitrary in representation. To do this, we must first consider the pure process of representation and the pure process of conceptualization. I need not say that when a person imagines a triangle, they can always imagine only this or that particular triangle. We must now consider the nature of the connection between mere representations and the nature of the connection between pure concepts. What governs our life of representation? When we have the representation of a rose, the representation of a person who gave us a rose may arise quite spontaneously. This may be followed by the image of a blue dress that the person in question was wearing, and so on. Such connections are called “associations of images.” But this is only one way in which people link images together. It occurs in its purest form when a person completely surrenders to the life of the imagination. But images can also be strung together according to other principles. Let this be illustrated with an example: A boy is sitting in the forest beneath tall trees. A man comes along the path and admires the fine timber. “Good morning, carpenter,” says the lively boy. Another person comes along and admires the bark. “Good morning, tanner,” says the quick-witted boy. A third person passes by and admires the magnificent growth of the trees. “Good morning, painter,” says the boy. —Here, then, three people see the same thing—the trees—and ideas arise in each of these three people, though they differ between the carpenter, the tanner, and the painter. These are different ways of linking ideas together, not mere associations. This is because people, according to their inner nature—the structure of their soul—connect this or that external idea with another; they do not merely surrender themselves externally to the ideas. Here, people allow the power that rises from within them to take effect. This is called apperception at work within them. Apperception and association are the forces that link mere ideas together through external or subjective, inner motivations. Both apperception and association operate in the realm of mere ideas. The situation is entirely different in the realm of concepts. Where would people end up if, in the realm of concepts, they were guided solely by the subject’s apperception and by chance association? Here, people must be guided by very specific laws that are independent of the association of ideas and of the subject’s apperception. If we focus solely on the mere external connection, we do not find the internal coherence of concepts. There is an internal coherence of concepts, and we find the laws governing this in formal logic.

[ 15 ] First, we must now examine the connection between two concepts. We connect the concept of a horse with that of running when we say: “The horse runs.” - We call such a connection between concepts a “judgment.” The point is that this connection must be made in such a way that only true judgments can result. Here we have, for the time being, merely a connection between two concepts, entirely independent of association and apperception. When we link two concepts through their content, we form a judgment. An association is not a judgment, for one could, for example, also link a bull and a horse together through such an association. However, the combination of concepts can also occur in more complex ways. We can link one judgment to another and thus arrive at a “conclusion.” A famous old example of this is the following: All humans are mortal. Cajus is a human. Therefore, Cajus is mortal. - Two judgments in these sentences are true, so the third one inferred from them—“Cajus is mortal”—is also true. A judgment is the combination of two terms: a subject and a predicate. When two judgments are combined and a third follows from them, this is an inference. We can now formulate a general pattern for this: If “Cajus” is the subject \(S\) and “mortal” is the predicate \(P\), then in the judgment “Cajus is mortal” we have the connection between the subject \(S\) and the predicate \(P: S = P\). Following this schema, we can form thousands of judgments. However, to arrive at a conclusion, we must also have a middle term \(M\)—in our example, “human” or “all humans.” Thus, for a conclusion, we can set up the schema as follows:

\(M = P\) All humans are mortal
\(S = M\) Cajus is a human
\(S = P\) Therefore, Cajus is mortal

[ 16 ] If this conclusion is to be correct, the concepts must be connected to one another in exactly this way; nothing must ever be transposed. For example, let us form the following sequence of judgments: The portrait resembles a person—The portrait is a work of art—we must not then conclude: Therefore, the work of art resembles a person. This latter conclusion would be false. What, then, is the source of the error here? We would have the following schema:

\(M = P\) The portrait resembles a person
\(M = S\) The portrait is a work of art
But \(S\) is not equal to \(P\):
The work of art does not resemble a person

[ 17 ] Here we have reversed the generally accepted schema. It therefore depends on the form of the schema, on the manner of the connection, to know that the first inferential figure is correct and the second is incorrect. It does not matter how the connection between the concepts otherwise takes place in our thoughts; it must be like the first formula in order to be correct.

[ 18 ] We will now see how to recognize a certain logical pattern in order to identify a number of such figures of speech. Correct thinking follows very specific figures of speech; otherwise, it is simply incorrect thinking. However, things are not always as straightforward as in this example. Simply based on the fact that the figures of speech are incorrect, one could often deduce today—even from the most scholarly books—that what is being said cannot be true. Thus, there are inner laws of thought just as there are laws of mathematics; one might call it an “arithmetic of thought.” Now you can imagine the ideal picture of correct thinking: all concepts must be formed according to the laws of formal logic. Formal logic, however, has certain limits. These limits must be applied to human intellectual life. In this way, one would arrive at correct insights and recognize the nature of fallacies. According to all the rules of logic, it would correspond to the laws of logic if we were to say:

All Cretans are liars \(M = P\)
This man is a Cretan \(S = M\)
Therefore, he is a liar, so \(S = P\)

[ 19 ] Now, the ancient logicians had already observed that this is true in all cases, except when a Cretan himself says it. In that case, the conclusion is certainly false. For if a Cretan says, “All Cretans are liars, therefore I am a liar”—then it would not be true that Cretans are liars, and thus he would be telling the truth; and so on.

[ 20 ] The same is true of all fallacies, for example, the so-called “crocodile fallacy”: An Egyptian woman saw her child, who was playing by the Nile, being seized by a crocodile. At the mother’s request, the crocodile promises to return the child if she can guess what it is about to do. The mother then says: “You will not return my child to me.” The crocodile replies: “Whether you spoke the truth or not, I will not return the child. For if what you said is true, you will not get it back, according to your own words. But if it is false, I will not return it according to our agreement.”—The mother: “Whether I spoke true or false, you must return my child to me. For if my statement is true, you must give it to me according to our agreement; but if it is false, then the opposite must be true. You will return my child to me.” The same applies to the conclusion concerning a teacher and a student. The teacher taught the student the law. The student is to pay the second half of the fee only after he has won his first lawsuit. After completing his studies, the student delays beginning his legal practice and, consequently, making the payment. Finally, the teacher sues him, saying: “Foolish youth! In any case, you must pay now.” For if I win the case, you must pay according to the court’s ruling; if you win, you must pay according to the contract, since you have won your first legal dispute.” — But the student replied: “Wise teacher! Under no circumstances do I need to pay. For if the judges rule in my favor, I have nothing to pay according to the court’s ruling; but if they rule against me, I pay nothing according to our contract.”

[ 21 ] There are countless fallacies of this kind that are formally correct. The point is that logic is applicable to everything except itself. The moment one turns to the subject itself, formal logic dissolves. This is a reflection of something else: When we move from the three bodies of the human being to the “I,” everything changes. The “I” is the arena of logic, which, however, may only be applied to other things, not to itself. No experience can ever be gained through logic; rather, logic can only bring order to experiences.